Stat 5101 Lecture Notes - School of Statistics

2.5. PROBABILITY THEORY AS LINEAR ALGEBRA 71Corollary 2.37. Every bounded random variable is in L 1 .Corollary 2.38. In a probability model with a finite sample space, every randomvariable is in L 1 .The corollaries take care of the trivial cases. Thus the question of existenceor non-existence of expectations only applies to unbounded random variablesin probability models on infinite sample spaces. Then Theorem 2.36 is usedto determine whether expectations exist. An expectation is an infinite sum inthe discrete case or an integral in the continuous case. The question is whetherthe integral or sum converges absolutely. That is, if we are interested in theexpectation of the random variable Y = g(X) where X has density f, we needto test the integral∫E(|Y |) = |g(x)|f(x)dxfor finiteness in the continuous case, and we need to test the corresponding sumE(|Y |) = ∑ x∈S|g(x)|f(x)for finiteness in the discrete case. The fact that the integrand or summand hasthe particular product form |g(x)|f(x) is irrelevant. What we need to know hereare the rules for determining when an integral or infinite sum is finite.We will cover the rules for integrals first. The rules for sums are very analogous.Since we are only interested in nonnegative integrands, we can alwaystreat the integral as representing “area under the curve” where the curve inquestion is the graph of the integrand. Any part of the region under the curvethat fits in a finite rectangle is, of course, finite. So the only way the area underthe curve can be infinite is if part of the region does not fit in a finite rectangle:either the integrand has a singularity (a point where it goes to infinity), or thedomain of integration is an unbounded interval. It helps if we focus on eachproblem separately: we test whether integrals over neighborhoods of singularitiesare finite, and we test whether integrals over unbounded intervals are finite.Integrals over bounded intervals not containing singularities do not need to bechecked at all.For example, suppose we want to test whether∫ ∞0h(x) dxis finite, and suppose that the only singularity of h is at zero. For any numbersa and b such that 0